3 2k-p Fractional Factorial DesignsLarge number of factors) large number of experiments) full factorial design too expensive) Use a fractional factorial design2k-p design allows analyzing k factors with only 2k-p experiments.2k-1 design requires only half as many experiments2k-2 design requires only one quarter of the experiments

5 Fractional Design FeaturesFull factorial design is easy to analyze due to orthogonality of sign vectors.Fractional factorial designs also use orthogonal vectors. That is:The sum of each column is zero.åi xij =0 8 jjth variable, ith experiment.The sum of the products of any two columns is zero.åi xijxil=0 8 j¹ lThe sum of the squares of each column is 27-4, that is, 8.åi xij2 = 8 8 j

6 Analysis of Fractional Factorial DesignsModel:Effects can be computed using inner products.

7 Example 19.1Factors A through G explain 37.26%, 4.74%, 43.40%, 6.75%, 0%, 8.06%, and 0.03% of variation, respectively. Use only factors C and A for further experimentation.

8 Sign Table for a 2k-p DesignSteps:Prepare a sign table for a full factorial design with k-p factors.Mark the first column I.Mark the next k-p columns with the k-p factors.Of the (2k-p-k-p-1) columns on the right, choose p columns and mark them with the p factors which were not chosen in step 1.

14 Other Fractional Factorial DesignsA fractional factorial design is not unique. 2p different designs.Confoundings:Not as good as the previous design.

15 Algebra of ConfoundingGiven just one confounding, it is possible to list all other confoundings.Rules:I is treated as unity.Any term with a power of 2 is erased.Multiplying both sides by A:Multiplying both sides by B, C, D, and AB:

19 Design Resolution Order of an effect = Number of termsOrder of ABCD = 4, order of I = 0.Order of a confounding = Sum of order of two termsE.g., AB=CDE is of order 5.Resolution of a Design= Minimum of orders of confoundingsNotation: RIII = Resolution-III = 2k-pIIIExample 1: I=ABCD  RIV = Resolution-IV = 24-1IV

24 Case Study 19.2: ConclusionsFor word processing throughput (TW): A (Preemption), B (Time slice), and AB are important.For interactive jobs: E (Fairness), A (preemption), BE, and B (time slice).For background jobs: A (Preemption), AB, B (Time slice), E (Fairness).May use different policies for different classes of workloads.Factor C (queue assignment) or any of its interaction do not have any significant impact on the throughput.Factor D (Requiring) is not effective.Preemption (A) impacts all workloads significantly.Time slice (B) impacts less than preemption.Fairness (E) is important for interactive jobs and slightly important for background jobs.

25 SummaryFractional factorial designs allow a large number of variables to be analyzed with a small number of experimentsMany effects and interactions are confoundedThe resolution of a design is the sum of the order of confounded effectsA design with higher resolution is considered better

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